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What's the meaning of without loss of generality in the cryptography (Zero Knowledge Proof)?

Without loss of generality, suppose we want to check if a 1 = a 2 . In the following description, j ∈ { 1, 2 } .

Reference: Zero-knowledge test of vector equivalence granulation of user data with privacy.

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    $\begingroup$ I’m voting to close this question because it's about mathematics in general, with no specific nuance in cryptography. Explanations are easy to find, e.g. Wikipedia, Math.SE. If you have a specific question about a specific proof about cryptography, feel free to ask here. But you need to ask a specific question and post the relevant part of the proof and its context (not just a screenshot). $\endgroup$ Commented Mar 5, 2022 at 15:06

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While this is better suited for Math.SE as suggested in the comment, it's an easy answer, so I'll just write it here.

It basically means, we're going to make a choice here, but the choice doesn't matter - you could replace this choice with any other choice, and the proof would work identically.

The second use of the phrase is the easiest to see. They assume $T_1$ is corrupted. This could, therefore, look like the proof only works for that one case, and doesn't work if $T_2$ was corrupted (or any other $T_i$). So they explicitly write that this choice does not lose generality, because you could replace $T_1$ with any other $T_i$ and the proof would still be identical.

I haven't checked the reference but I guess the first "WLOG" means $a_1 = a_2$ could be replaced with $b_1 = b_2$ for an identical proof.

Another common scenario this is used in, is if you have two variables $a, b$, then you could say "without loss of generality, we assume $a\geq b$. We don't lost generality because we could just re-label the two variables.

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